The Stacks project

Lemma 59.86.2. Consider a tower of cartesian diagrams of schemes

\[ \xymatrix{ W \ar[d]_ i & Z \ar[l]^ j \ar[d]^ k \\ X \ar[d]_ f & Y \ar[l]^ h \ar[d]^ e \\ S & T \ar[l]_ g } \]

Let $K$ in $D(T_{\acute{e}tale})$. If

\[ f^{-1}Rg_*K \to Rh_*e^{-1}K \quad \text{and}\quad i^{-1}Rh_*e^{-1}K \to Rj_*k^{-1}e^{-1}K \]

are isomorphisms, then $(f \circ i)^{-1}Rg_*K \to Rj_*(e \circ k)^{-1}K$ is an isomorphism. Similarly, if $\mathcal{F}$ is an abelian sheaf on $T_{\acute{e}tale}$ and if

\[ f^{-1}R^ qg_*\mathcal{F} \to R^ qh_*e^{-1}\mathcal{F} \quad \text{and}\quad i^{-1}R^ qh_*e^{-1}\mathcal{F} \to R^ qj_*k^{-1}e^{-1}\mathcal{F} \]

are isomorphisms, then $(f \circ i)^{-1}R^ qg_*\mathcal{F} \to R^ qj_*(e \circ k)^{-1}\mathcal{F}$ is an isomorphism.

Proof. This is formal, provided one checks that the composition of these base change maps is the base change maps for the outer rectangle, see Cohomology on Sites, Remark 21.19.5. $\square$

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